[FOM] BOOLE AND THE EMPTY SET
John Corcoran
corcoran at buffalo.edu
Wed Jun 30 17:52:44 EDT 2004
0.1 "The" empty class: Was the empty class mentioned or discussed before
Boole 1847, 21? What is the accepted history of the empty class? And
where do we find it? Wasn't there a French person who wrote about this
in English in the 1980s? Kanamori 2003,274 (BSL Volume 9 Number 3) says:
"The work of George Boole was a cresting of extensionalism in the 19th
century. He introduced "0" without explanation in his The Mathematical
Analysis of Logic [1847:21], and used it forthwith as an "elective
symbol" complementary to his "1" denoting the "Universe"."
0.2 Should Boole deserve the credit for bringing the empty class to our
attention? I have not seen this point made or contradicted by a serious
historian. Curiously, Kneale and Kneale 1962 do not mention the null
class until they get to Boole, page 407UP14 in my 1964 edition. I would
have guessed Leibniz, Euler, or even De Morgan 1846.
0.3Does Boole 1847 ever refer to "0" as an elective symbol? This strikes
me as a mistake since elective symbols represent operations not classes.
What are the various names Boole uses for zero, the number, and what
does he say about it? What are the various names Boole uses for "the"
empty class, the entity, and what does he say about it? Does it exist
in the same sense that the universe exists? 0.4 Is there a constant
elective operation referred to in 1847 which yields the empty class for
each argument?
0.5 What evidence do we have that Boole discovered the empty set by
analogy? Was it after concluding that the expressions of analysis must
admit of interpretation in the realm of classes that Boole asked how "0"
MUST be interpreted? Or was the affinity of zero with the empty class
one of the thoughts that led him to the idea that the expressions of
analysis MUST be interpretable in the realm of classes? Where is this
discussed?
0.6 Is there any difference between Boole's treatment of the null set in
1847 and his treatment of it in later works 1848 or 1854?
0.7 Who were the early skeptics to dispute the existence, or the
coherence, of the null set?
0.8 Among the serious logicians who paid attention to history [ Beth,
Church, Kleene, Lukasiewicz, Tarski, Skolem] was there any discussion of
the history of the concept of the null set?


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